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On the rate of convergence of probabilities of moderate deviations

Published online by Cambridge University Press:  09 April 2009

V. K. Rohatgi
Affiliation:
The Catholic University of AmericaWashington, D.C.
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Let {Xn: n ≧ 1} be a sequence of independent random variables and write Letand let . Suppose that converges in law to the standard normal distribution (see [5, 280] for necessary and sufficient conditions). Let {xn} be a monotonic sequence of positive real numbers such that xn → ∞ as n → ∞. Then as n → ∞ for all ε > 0. [6] Rubin and Sethuraman call probabilities of the form probabilities of moderate deviations and obtain asymptotic forms for such probabilities under appropriate moment conditions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Baum, L. E., Katz, M. L., and Read, R. R., ‘Exponential convergence rates for the law of large numbers’, Trans. Amer. Math. Soc., 102 (1962), 187199.Google Scholar
[2]Davis, James Avery, ‘Convergence rates for probabilities of moderate deviations’, Department of Mathematics and Statistics. The University of New Mexico, Tech. Report No. 147, (1967).Google Scholar
[3]Feller, W., An Introduction to Probability Theory and its Applications, Vol. II. (Wiley, New York, 1966).Google Scholar
[4]Heyde, C. C., and Rohatgi, V. K., ‘A pair of domplementary theorems on convergence rates in the law of large numbers’, Proc. Cambr. Phil. Soc., 63 (1967), 7382.Google Scholar
[5]Loève, M., Probability Theory. (Van Nostrand, 3rd Ed., New York, 1963).Google Scholar
[6]Rubin, H., and Sethuraman, J., ‘Probabilities of moderate deviations’, Sankhya, 27 (1965) Series A, 325346.Google Scholar